Question

Quadratic functions can be used to describe the rate at which an object will fall and predict when it will land or how high it may go. Create a quadratic function using the function y=-16t^2 to help predict when an object will return to the ground from being kicked, dropped, or thrown.
What does the 10 in my equation represent?

Answer 1

In the context of the quadratic function \( y = -16t^2 + h \), where \( h \) is the initial height from which the object is thrown (or kicked or dropped), the term \( 10 \) would represent the initial height \( h \) (in feet) from which the object is launched.

So, if you have the equation \( y = -16t^2 + 10 \), the \( 10 \) indicates that the object is initially 10 feet above the ground when it is kicked or thrown. The function describes how the height \( y \) of the object changes over time \( t \), where the term \( -16t^2 \) represents the effect of gravity (specifically, the acceleration due to gravity in feet per second squared).

To predict when the object will return to the ground, you would set \( y = 0 \) (when the object hits the ground) and solve for \( t \):

\[
0 = -16t^2 + 10
\]

Rearranging the equation gives:

\[
16t^2 = 10
\]

\[
t^2 = \frac{10}{16} = \frac{5}{8}
\]

\[
t = \sqrt{\frac{5}{8}} \approx 0.79 \text{ seconds}
\]

So, it would take approximately \( 0.79 \) seconds for the object to return to the ground if it was kicked from an initial height of 10 feet.